展开 (x-1)^3(1/x+x)^6

解答

展开

(x - 1)^{3} (\frac{1}{x} + x)^{6}

\frac{( x ^{2} + 1 ) ^{6}}{x ^{3}} - \frac{3 ( x ^{2} + 1 ) ^{6}}{x ^{4}} + \frac{3 ( x ^{2} + 1 ) ^{6}}{x ^{5}} - \frac{( x ^{2} + 1 ) ^{6}}{x ^{6}}

求解步骤

(x - 1)^{3} (\frac{1}{x} + x)^{6}

展开

(x - 1)^{3} : x^{3} - 3 x^{2} + 3 x - 1

= (x^{3} - 3 x^{2} + 3 x - 1) (\frac{1}{x} + x)^{6}

= (\frac{1}{x} + x)^{6} (x^{3} - 3 x^{2} + 3 x - 1)

打开括号

= (\frac{1}{x} + x)^{6} x^{3} + (\frac{1}{x} + x)^{6} (- 3 x^{2}) + (\frac{1}{x} + x)^{6} \cdot 3 x + (\frac{1}{x} + x)^{6} (- 1)

化简

(\frac{1}{x} + x)^{6} x^{3} + (\frac{1}{x} + x)^{6} (- 3 x^{2}) + (\frac{1}{x} + x)^{6} \cdot 3 x + (\frac{1}{x} + x)^{6} (- 1) : \frac{( x ^{2} + 1 ) ^{6}}{x ^{3}} - \frac{3 ( x ^{2} + 1 ) ^{6}}{x ^{4}} + \frac{3 ( x ^{2} + 1 ) ^{6}}{x ^{5}} - \frac{( x ^{2} + 1 ) ^{6}}{x ^{6}}

= \frac{( x ^{2} + 1 ) ^{6}}{x ^{3}} - \frac{3 ( x ^{2} + 1 ) ^{6}}{x ^{4}} + \frac{3 ( x ^{2} + 1 ) ^{6}}{x ^{5}} - \frac{( x ^{2} + 1 ) ^{6}}{x ^{6}}

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